Case 03 · route optimization · Medium · About 8–12 minutes
A messenger must collect a seal from every island station before reaching the ship. The footbridges are open in both directions, but the tide leaves little time for detours.
Your task is to plan one complete circuit through the stations. The illustration sets the scene; the bridge table is the actual map.
Start at Harbor H and finish at Ship Station S. Visit Orchard O, Mill M, Tower T, and Quarry Q exactly once each. Do not revisit H or any other station.
Only these bridges exist. Each can be crossed in either direction, and the stated travel times are exact.
| Bridge | Minutes |
|---|---|
| H–O | 4 |
| H–M | 6 |
| O–M | 2 |
| O–T | 7 |
| M–Q | 3 |
| M–T | 4 |
| Q–T | 2 |
| Q–S | 8 |
| T–S | 5 |
Collecting a seal takes no extra time. You must finish at S only after visiting all four middle stations.
What is the fastest valid route, and how many minutes does it take? A quick route that misses one station is not valid. Nor may you invent a bridge because two places look close in the artwork.
Hint 1 - reveal when you want a nudge
S can only be approached from Q or T. H can only be left toward O or M. Use those restricted ends to organize possible routes.
Hint 2 - reveal when you want a nudge
There are only four complete routes through all stations. Compare H–O–M–T–Q–S with H–O–M–Q–T–S first.
Reveal the solution and explanation
The fastest route is H–O–M–Q–T–S, taking 16 minutes.
Add its bridge times: 4 + 2 + 3 + 2 + 5 = 16.
The complete list of valid routes is short. H–O–M–T–Q–S takes 20 minutes. H–O–M–Q–T–S takes 16. H–O–T–M–Q–S takes 26. H–M–O–T–Q–S takes 25.
Every other ordering of O, M, T, Q requires a bridge absent from the table. For example, a route cannot leave O directly for Q. The four middle stations have 24 possible orders, and only the four listed orders survive the connection test.
To repeat the check by hand, list the possible four-letter middle orders. Cross out an order as soon as two consecutive stations lack a bridge. Add travel times only for complete survivors; a missing connection cannot be repaired by guessing a shortcut.
Comparing their totals proves that 16 minutes is the minimum. No tie remains, so both the route and its travel time are determined.
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